CAMBA Cleveland Area Mountain Bike Association is the local bike club in that area. You can also monitor trail conditions when the time comes on Ohiosingletrack.com Forums
Closest places would be Reagan & Hoffman City Parks in Midina, Findley State Park, and Royalveiw in Cleveland Metro Parks.
Maybe someone else can give you some more ideas?

I'm a CAMBA member. PM me and I will try to get you in touch with somebody if you want a guide or a group ride.
Findley is probably closest to Lorain. It will be dry and fast by July. I work right down the street from it.

Royalview is much like Findley, but with a little elevation, better scenery, and a new tech. section opening soon with skinnies, rocks, logs... There may be weekly group rides happening there in July.

Most trails around Cleveland/Lorain are pretty flat, but there is actually some variety.

Ohio & Erie Canal trail is like a roller coaster. It is very short, but very fun. There's a video of it on the CAMBA front page. It is connected to a flat bike path that runs along the river into the Cuyahoga Valley Nat'l Park (no cars - just a few road crossings).

As you get farther from Lake Erie, you get more hills. West Branch SP and Mohican SP are def. worth the drive if you are up for more mileage and challenge.